This article deals with the cake cutting problem. In this setting, there exists two notions of fair division: proportional division (when there are n players, each player thinks to get at least 1/n of the cake) and envy-free division (each player wants to keep his or her share because he or she does not envy the portion given to another player). Some results are valid for proportional division but not for envy-free division. Here, we introduce and study a scale between the proportional division and the envy-free division. The goal is to understand where is the gap between statements about proportional division and envy-free division. This scale comes from the notion introduced in this article: k-proportionality. When k = n this notion corresponds to the proportional division and when k = 2 it corresponds to envy-free division. With k-proportionality we can understand where some difficulties in fair division lie. First, we show that there are situations in which there is no k-proportional and equitable division of a pie with connected pieces when k $\le$ n -1. This result was known only for envy-free division, ie k = 2. Next, we prove that there are situations in which there is no Pareto-optimal k-proportional division of a cake with connected pieces when k $\le$ n -1. This result was known only for k = 2. These theorems say that we can get an impossibility result even if we do not consider an envy-free division but a weaker notion. Finally, k-proportionality allows to give a generalization with a uniform statement of theorems about strong envy-free and strong proportional divisions.

We consider the problem of fair division of a two dimensional heterogeneous good among several agents. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols either consider a one-dimensional resource, or allocate each agent several disconnected pieces. In practice, however, the two dimensional shape of the allotted piece is of crucial importance in many applications, e.g., squares or bounded aspect-ratio rectangles are most useful for building houses as well as advertisements. We thus introduce and study the problem of envy-free two-dimensional division wherein the utility of the agents depends on the geometric shape of the allocated pieces (as well as the location and size). In addition to envy-freeness, we require that the fraction allocated to each agent be at least a certain constant that depends only on the shape of the cake and the number of agents. We focus on the case where the allotted pieces must be square and the cakes are either squares or the unbounded plane. We provide algorithms for the problem for settings with two and three agents.